# Tensor product

#### Tensor product of two unitary modules

The tensor product of two unitary modules and over an associative commutative ring with a unit is the -module together with an -bilinear mapping

which is universal in the following sense: For any -bilinear mapping , where is an arbitrary -module, there is a unique -linear mapping such that

The tensor product is uniquely defined up to a natural isomorphism. It always exists and can be constructed as the quotient module of the free -module generated by the set modulo the -submodule generated by the elements of the form

then . If one gives up the requirement of commutativity of , a construction close to the one described above allows one to form from a right -module and a left -module an Abelian group , also called the tensor product of these modules [1]. In what follows will be assumed to be commutative.

The tensor product has the following properties:

for any -modules , and .

If and are bases of the free -modules and , then is a basis of the module . In particular,

if the are free finitely-generated modules (for instance, finite-dimensional vector spaces over a field ). The tensor product of cyclic -modules is computed by the formula

where and are ideals in .

One also defines the tensor product of arbitrary (not necessarily finite) families of -modules. The tensor product

is called the -th tensor power of the -module ; its elements are the contravariant tensors (cf. Tensor on a vector space) of degree on .

To any pair of homomorphisms of -modules , , corresponds their tensor product , which is a homomorphism of -modules and is defined by the formula

This operation can also be extended to arbitrary families of homomorphisms and has functorial properties (see Module). It defines a homomorphism of -modules

which is an isomorphism if all the and are free and finitely generated.

An important interpretation of the tensor product in (theoretical) physics is as follows. Often the states of an object, say, a particle, are defined as the vector space over of all complex linear combinations of a set of pure states , . Let the pure states of a second similar object be , , yielding a second vector space . Then the pure states of the ordered pair of objects are all pairs and the space of states of this ordered pair is the tensor product .

#### Tensor product of two algebras

The tensor product of two algebras and over an associative commutative ring with a unit is the algebra over which is obtained by introducing on the tensor product of -modules a multiplication according to the formula

This definition can be extended to the case of an arbitrary family of factors. The tensor product is associative and commutative and contains a unit if both algebras have a unit. If and are algebras with a unit over the field , then and are subalgebras of which are isomorphic to and and commute elementwise. Conversely, let be an algebra with a unit over the field , and let and be subalgebras of it containing its unit and such that for any . Then there is a homomorphism of -algebras such that , . For to be an isomorphism it is necessary and sufficient that there is in a basis over which is also a basis of the right -module .

#### Tensor product of two matrices (by D.A. Suprunenko)

The tensor product, or Kronecker product (cf. Matrix multiplication), of two matrices and is the matrix

Here, is an -matrix, is a -matrix and is an -matrix over an associative commutative ring with a unit.

Properties of the tensor product of matrices are:

where ,

If and , then

Let be a field, and . Then is similar to , and , where is the unit matrix, coincides with the resultant of the characteristic polynomials of and .

If and are homomorphisms of unitary free finitely-generated -modules and and are their matrices in certain bases, then is the matrix of the homomorphism in the basis consisting of the tensor products of the basis vectors.

#### Tensor product of two representations (by A.I. Shtern)

The tensor product of two representations and of a group in vector spaces and , respectively, is the representation of in uniquely defined by the condition

 (*)

for all , and . If and are continuous unitary representations of a topological group in Hilbert spaces and , respectively, then the operators , , in the vector space admit a unique extension by continuity to continuous linear operators , , in the Hilbert space (being the completion of the space with respect to the scalar product defined by the formula ) and the mapping , , is a continuous unitary representation of the group in the Hilbert space , called the tensor product of the unitary representations and . The representations and are equivalent (unitarily equivalent if and are unitary). The operation of tensor multiplication can be defined also for continuous representations of a topological group in topological vector spaces of a general form.

If is a representation of an algebra in a vector space , , one defines the tensor product , which is a representation of in , by

In case is a bi-algebra (cf. Hopf algebra), composition of this representation with the comultiplication (which is an algebra homomorphism) yields a new representation of , (also) called the tensor product.

In case is a group, a representation of is the same as a representation of the group algebra of , which is a bi-algebra, so that the previous construction applies, giving the same definition as (*) above. (The comultiplication on is given by .)

In case is a Lie algebra, a representation of is the same as a representation of its universal enveloping algebra, , which is also a bi-algebra (with comultiplication defined by , ). This permits one to define the tensor product of two representations of a Lie algebra:

#### Tensor product of two vector bundles

The tensor product of two vector bundles and over a topological space is the vector bundle over whose fibre at a point is the tensor product of the fibres . The tensor product can be defined as the bundle whose transfer function is the tensor product of the transfer functions of the bundles and in the same trivializing covering (see Tensor product of matrices, above).

 [1] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) [2] F. Kasch, "Modules and rings" , Acad. Press (1982) (Translated from German) [3] A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian) [4] P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958) [5] M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967)